Does the "jetpack radius" of a finite subset of a metric space have an accepted name?

Question

Definition 0. Whenever $X$ is a metric space and $A \subseteq X$ is a finite set, define that the jetpack radius of $A$ is the minimum among all $r \in \mathbb{R}_{\geq 0}$ such that if we draw an edge between any two elements of $A$ that are distance at most $r$ apart, then the we get a connected graph.

So intuitively, the jetpack radius of $A$ is how far we have to be able to jump per use of our jetpack, if we want to be able to get from one vertex to any other vertex.

This notion can be defined more generally for a finite metric space.

Definition 1. Whenever $A$ is a finite metric space, define that the jetpack radius of $A$ is the minimum among all $r \in \mathbb{R}_{\geq 0}$ such that if we draw an edge between any two elements of $A$ that are distance at most $r$ apart, then the we get a connected graph.

We recover Definition 0 by studying the metric on $A$ obtained by restriction. Anyway, I'd like to learn the basic facts about these numbers.

Question. Does the "jetpack radius" of a finite subset of a metric space have an accepted name, and is it discussed anywhere?

Answer 1

Let's denote the "jetpack radius" $JR$ for now.

The graph you describe is sometimes called the Rips graph; it is the $1$-skeleton of the Vietoris–Rips complex. In applied literature on data clustering it is called the $\epsilon$-neighborhood graph.

A metric space with $JR=0$ is called chain connected: sample reference.

A metric space with $JR < \infty$ is called coarsely connected: sample reference.

It is easy to see that $JR = \sup_{B\subset A}\operatorname{dist}( B, A\setminus B)$; that is, the maximal possible distance between two complementary subsets of $A$. In cluster analysis, the search for this partition is called single-linkage clustering with $2$ clusters.

Yet another interpretation: $JR/2$ is the Gromov–Hausdorff distance from $A$ to the set of all connected metric spaces; that is $$JR/2 = \inf\{d_{GH}(A, Y) : Y \text{ is a connected metric space}\} $$

In IEEE papers on wireless communication, they call it the critical transmission radius for connectivity, since it's the transmission radius that devices must have in order for their network to remain connected.

The terms attached to $JR$, usually in its stochastic version (connected with high probability) are (sample reference):

• critical radius for connectivity
• critical connectivity radius
• connectivity threshold

The notation $r_{\textrm{con}}$ is sometimes used.